Fourier Transform Analysis: (1) X-ray Diffraction Effects by Finite Montmorillonite and Mica Crystals

Malcolm Ross
U.S. Geological Survey, Washington, D.C. 20242

Abstract: A computer program has been developed to generate the X-ray diffraction intensity distribution along any particular reciprocal lattice row, plane, or volume, for any arbitrary group of atoms within a crystal. The program, which maps the intensity in crystal reciprocal space in much the same way as a conventional Fourier series program maps the electron density in direct crystal space, has been used to calculate the expected X-ray diffraction line profiles for a number of montmorillonite and mica crystallites of varying thicknesses in the c* direction.

The program evaluates the function G(HKL)= ∑ n−1 N f n  exp⁡ 2ni(H x n +K y n +L z n ), where G(HKL) is the Fourier transform of an array of N-atoms at a particular H, K, L coordinate in reciprocal space, fn is the scattering factor of the nth atom, and x, y, z its coordinates in direct space. The function is evaluated for all N-atoms within the finite model crystal under study for non-integral as well as integral values of H, K, and L. In practice a complete line profile is made by calculating G(HKL) at intervals in the range of (100 Å)−1 to (10,000 Å)−1.

The apparent d-spacings of the various clay mineral models, as given by the line profiles, approach asymptotically the true value as the number of layers increase. For example, the apparent d001 spacing for a mica of the composition K (Fe, Mg) 3 Si 3 Al 10 (OH) 2 is 12.91, 11.35, 10.79, 10.53, 10.38, 10.22, 10.14, 10.04 and 10.02 Å for crystals 2, 3, 4, 5, 6, 8, 10, 20, and 30 layers thick, respectively. For the infinitely thick crystal, d001=10.000 Å. The apparent d001 spacing for a montmorillonite of the composition K0·33Al2(Si, Al)4O10(OH)2·4H2O (true d001=15.400 Å) is 18.85, 16.80, 15.87, 15.52, and 15.41 Å for crystals 2, 3, 5, 10, and 30 layers thick, respectively.

These diffraction profiles and line shifts can be used in analyzing montmorillonites, micas, and mixed-layer montmorillonite-mica clays.

Clays and Clay Minerals; 1967 v. 15; no. 1; p. 47; DOI: 10.1346/CCMN.1967.0150105
© 1967, The Clay Minerals Society
Clay Minerals Society (www.clays.org)